Theory-Crafting: "Crossing the T"

[tortugapower]

Hi all,

I've been thinking a lot about the concept of "crossing the T" lately. In particular, I wonder if it's useful in the modern (let's say 1930s onward) battlefield.  I think of it mostly as an Age of Sail thing, but there have been a persistent set of folks commenting to me saying that crossing the T has advantages that persist into modern times.

I want to focus in particular on the idea that "splash-down pattern of shells is an ellipse, therefore a ship with a bow/stern aspect is easier to hit". If this is true, it means that crossing the enemy's T should make their ships easier to hit.

However, first I want to put down what I think are the pros/cons in the "modern" era because I'm sure experts here can help me fill the list.

"Crossing the T" pros:
- ships in a line [astern] formation cannot fire, asides from leader
- crossing ships get full broadside, crossed ships have forward-firing weapons only
- easier to hit (??)

"Crossing the T" cons:
- none?
- except: are crossed enemy ships perhaps harder to hit? (according to @jwsmith this may be how RtW models it)

Next post, I'll look into whether this elliptical splashdown means ships are harder to hit when abeam.

Cheers,
Tortuga

[tortugapower]

Theory Post #1: The ease/difficulty of hitting ships.

I will be modeling shell trajectories, using simple Newtonian physics and including drag.  I'm going to be using the Iowa-class battleship for all trajectory modeling because information about this ship is abundant.

Shell velocity: 762 m/s
Max elevation: 45 degrees

In order to calculate the drag coefficient simply, I find one that matches a maximum range of 25 miles.  I think I picked that up from a page about the Missouri, but if someone finds a different value for the max range of the 762 m/s shells, I'll be happy to redo my drag coefficient.

Anyway, I should just say: I don't care to be 100% precise, I just want good approximations of reality for my simulations.

I get a drag coefficient of 0.006 from this.  My blind guess was 0.01, so this number feels right. Fun note: maximum range is around ~41 degrees elevation with drag (instead of 45 degrees ideally).



Now, I want to investigate the notion that shell accuracy is much more impacted by the y-errors (elevation) than x-errors (turret rotation). First of all, I agree without evidence that there are more errors of a higher amplitude in the y-direction:
- ship roll/yaw from sea state impacts elevation
- gun firing can cause ship roll (but by only a small amount, I've been told)

I want to look at ranging accuracy (human error) in a different post.

Because of the above qualitative reasons, I assign an arbitary 5:1 ratio of vertical:horizontal errors in accuracy. For this next plot, I take the expected elevation and both add and subtract 0.1 degrees, vertically, and 0.02 degrees horizontally. I find the maximum spread of the shots within this range and call that my aiming error or inaccuracy in that axis.

---

Edit: FIGURE IS WRONG.

(Leaving for reference, but the above is wrong. I was using degrees instead of radians. Horizontal should be about 10x less spread.)

I'll have a more accurate plot in next theory post. For now, the below is mostly correct, except where struck-out.

---

It seems the effect of inaccuracy in the elevation (vertical precision) matters less for higher elevation.  This makes sense, too, because if you expect a fixed error in degrees, the relative error compared to the total elevation decreases as you raise your gun. Basically 5 +/- 0.1 degrees is a bigger relative spread of angles than 50 +/- 0.1 degrees.

Conversely, horizontal inaccuracy is just a coefficient of total distance: the farther your shell travels, the more error it accumulates due to aiming error horizontally.

The above plot should mean that the elliptical splashdown pattern is only oblong to favor the line towards the target when the range is close and elevation is low. The elliptical splashdown mostly favors a line towards the target, until almost maximum firing distance (see Theory Post #2).

The next thing I need to show, then, is that when the elevation is low (at shorter ranges), elevation error doesn't matter as much because of shell trajectory.

That will be in my next post.

[hiiiiii74]

An interesting idea, but some other factors may be worth considering before diving too deep into feasibility.

I think that something worth noting here is whether or not you are concerned about shell penetration upon impact. If you are at practical battle range, particularly pre-1930s or so, odds are, you are going to be making consistent contact with the enemy's vertical belt armor. In this scenario of 'Crossing the T', the enemy capital ship's belt armor is going to be at a very oblique angle to your shell's impact, because of the relative angles of the ships in question. At long ranges, plunging fire does not care much about the relative angle of the two ships, so it does not matter as much.

What works for tanks works for ships, because it's plain physics. If the enemy battleship is at a 60 degree angle relative to you (which will happen if you steam across their bow or stern), if they have 12" of belt armor, the effective thickness of that armor will actually become 24", if all other things remain equal. If you are broadside while doing so, your armor remains at its nominal thickness, while you have just gifted the enemy extra inches of armor for 0 displacement. If you are hoping for hits on turrets, or HE hits on the superstructure, perhaps this is not as big of a deal for you. Last I have heard, the game models the relative angles of the ships when the main belt armor is struck by a shell. I do not know if the game models deflection, because at the most extreme angles, armor that has the strength to hold up against the sheer kinetic force of the impact will send the shell off the armor and off into the nearby water to detonate as a near-miss, or simply detonate against the side of the ship, causing far less damage than a full penetration would.

In addition, you are making the fire control solution increasingly difficult. When you are perpendicular to the enemy, the 'derivative' (for lack of a better term) of the fire control solution is at a very high point, perhaps even the highest point. Perhaps heading in the opposite direction of a somewhat distant enemy would produce a more fluctuating fire control solution. When perpendicular to the enemy, their distance to you is increasing at a much higher rate than if you were steaming parallel. In addition, if you are moving at all, their bearing relative to you is also changing at a higher rate than if you were parallel to them. I believe the game also models these factors into your hit chance. I do know for sure that if your fire control solution has remained steady over the last few ranging barrages, your accuracy increases greatly.

Personally, I do not 'Cross the T' until the enemy has taken so much damage that I need to change course to stay within range, and can run circles around them. By that point, it's no longer a tactical maneuver as much as it is a clean up maneuver. I believe the maneuver fell out of style for a reason.

[Fredrik W]

Ships are harder to hit from ahead or astern. This is due to the smaller target area presented. You can se this in the hit chances in the game, there will be a modifier for "Target aspect". So, somewhat counter intuitively, ships are harder to hit when crossing the T, but on the other hand you will have more guns bearing than the enemy.

As for armor penetration, the game assumes that end bulkheads are the same thickness as the belt (this was not always true in real ships, but it is a reasonable assumption), so penetration from ahead is the same as if side on. BUT penetration does take angle into effect, so if the firing ship is at around 45 degrees angle penetration will be reduced for belt hits.

[dorn]

Crossing T is important if both sides has long line of ships. If you have just 2 or 3 ships, you can adjust course and quickly cancel advantage of crossing T before enemy get into range with reasonable accuracy.

[rimbecano]

Nov 1, 2019 19:19:19 GMT -6 tortugapower said:

The above plot should mean that the elliptical splashdown pattern is only oblong to favor the line towards the target when the range is close and elevation is low.

This assumes that range dispersion is mostly a function of elevation error, but I believe shot-to-shot variation in muzzle velocity is a significant contributor.

[mdesanta]

Nov 2, 2019 1:01:10 GMT -6 Fredrik W said:

As for armor penetration, the game assumes that end bulkheads are the same thickness as the belt (this was not always true in real ships, but it is a reasonable assumption), so penetration from ahead is the same as if side on. BUT penetration does take angle into effect, so if the firing ship is at around 45 degrees angle penetration will be reduced for belt hits.

Does the angle matter in terms of overpenetration? As in less chances of pass-through hits the more bow/stern-on the enemy is, as there's more of the ship to go through?

[mdesanta]

Nov 2, 2019 1:08:02 GMT -6 dorn said:

Crossing T is important if both sides has long line of ships. If you have just 2 or 3 ships, you can adjust course and quickly cancel advantage of crossing T before enemy get into range with reasonable accuracy.

But if I decide instead to plow ahead bow-on towards the enemy line with my all forward-mounted turrets layout (that design is to ships what bullpups are to guns, unaesthetic and revolting on a fundamental level) I should be absolutely punished by the wrath of god like the disgusting heretic I am.

[dorn]

Nov 2, 2019 6:04:31 GMT -6 mdesanta said:

Nov 2, 2019 1:08:02 GMT -6 dorn said:

Crossing T is important if both sides has long line of ships. If you have just 2 or 3 ships, you can adjust course and quickly cancel advantage of crossing T before enemy get into range with reasonable accuracy.

But if I decide instead to plow ahead bow-on towards the enemy line with my all forward-mounted turrets layout (that design is to ships what bullpups are to guns, unaesthetic and revolting on a fundamental level) I should be absolutely punished by the wrath of god like the disgusting heretic I am.

You are completely right but the only 1st class battleships used all forward turrets layout with ability to fire forward were Richelieu class build quite late. As I mentioned crossing T was less important later but still sometimes get decisive advantage even in WW2.

[tortugapower]

Nov 2, 2019 1:01:10 GMT -6 Fredrik W said:

Ships are harder to hit from ahead or astern. This is due to the smaller target area presented. You can se this in the hit chances in the game, there will be a modifier for "Target aspect". So, somewhat counter intuitively, ships are harder to hit when crossing the T, but on the other hand you will have more guns bearing than the enemy.

I keep getting reports of people who say that, because of the elliptical splashdown (larger spread in the direction of the shell travel), that even though the ship has a narrower head-on profile, it is easier to hit head-on because the axis of shell travel is more prone to error.

The oft-cited reference for this is the NavWeaps website:
www.navweaps.com/index_inro/INRO_BB-Gunnery.php

I disagree with this (so agree with RtW that head-on ships should be harder to hit), for reasons I'll show in an upcoming theory post. But did you model this on your intuition, Fredrik? Or can we chalk this up to the extensive research that williammiller (or another) has done?

[cv10]

Another downside for crossing the T is that it puts your BBs broadside to enemy destroyers

[williammiller]

There are several factors that affect accuracy when comparing a broadside to a end-on target....one aspect that is sometimes forgotten is the (rather important) 'relative speed' aspect. With a target that is moving directly towards/away from the firer it is more difficult to accurately assess target speed (unless you have decent RADAR), thus the relative speed part of the targeting formula will very likely have greater error, leading to greater 'range' error rates.

Rangefinders of the period are also going to have greater error in determining range to a target with the much more narrow target width (aspect) presented on a ends-on range estimate....this also can compound/add to the error rate in range estimation.

[Fredrik W]

Nov 2, 2019 10:19:13 GMT -6 tortugapower said:

Nov 2, 2019 1:01:10 GMT -6 Fredrik W said:

Ships are harder to hit from ahead or astern. This is due to the smaller target area presented. You can se this in the hit chances in the game, there will be a modifier for "Target aspect". So, somewhat counter intuitively, ships are harder to hit when crossing the T, but on the other hand you will have more guns bearing than the enemy.

I keep getting reports of people who say that, because of the elliptical splashdown (larger spread in the direction of the shell travel), that even though the ship has a narrower head-on profile, it is easier to hit head-on because the axis of shell travel is more prone to error.

The oft-cited reference for this is the NavWeaps website:
www.navweaps.com/index_inro/INRO_BB-Gunnery.php

I disagree with this (so agree with RtW that head-on ships should be harder to hit), for reasons I'll show in an upcoming theory post. But did you model this on your intuition, Fredrik? Or can we chalk this up to the extensive research that williammiller (or another) has done?

It was facts not intuition. Research by William and other sources all point to that being true.

[13th Fleet]

It's not just where the shell hits the water that matters, but also if the shell's flight path intersects the target ship. The height of a ship means it will catch shells that would've overshot a ship-shaped patched of water. A broadside engagement gives you a larger "shadow" on the water.

[srndacful]

IMHO, 'crossing the T' is nothing but a simple exercise in mathematics:
The side that's doing the crossing will be able to bring more guns to bear, and thus inflict more pain on the enemy - and since wars (and thus battles) are won by the side that inflicts the most pain on the enemy - the side that's doing the crossing will automatically win. 
The reduced chance of hitting is still offset by the additional number of guns able to fire on the enemy. 

Hell, let's make a mathematical model - Lanchester style:
Side A is 'crossing the T' on side B.
They both have 8 ships with guns in 4x2 configuration - however, Side A will, naturally, have all it's guns available - but Side B will only have 1/2 of it's guns initially available - so, basically a 2x2 configuration.
Let's take into account the reduced chance of hitting by giving Side A 10 guns to make a hit - while Side B needs only 8 guns to make a hit.
Each hit will take out a turret (with 2 guns in it) - so, basically, Side A will lose a ship in 4 hits, and Side B a ship with 2 hits.
Yes, I'm aware that Side B's ships will be a lot less damaged than Side A's, but their turrets will be out of action - and thus they will (basically) be sitting ducks - to be taken out later. 

With those rules in place - let's fight!

Round 1: 
Side A has 8x4x2 = 64 guns available = 6.4 hits - let's say 6, with the .4 carried over.
Side B has 8x2x2 = 32 guns available = 4 hits total.
So, basically, Side B has just lost 3 ships (5 ships remaining) - and Side A only 1 (7 remanining). 

Round 2: 
Side A has 7x4x2 = 56 guns available = 5.6 +0.4 (leftover from last round) = 6 hits
Side B has 5x2x2 = 20 guns available = 2.5 hits - so, 2 hits, with 0.5 carried over. 
So, Side B again loses 3 ships (2 remaining) - while Side A loses 0.5 ships (let's say damaged) 

Round 3:
Side A has 6.5x4x2 = 52 guns available = 5.2 hits & more than enough to wipe out the enemy.
Side B has 2x2x2 = 8 guns available = 1 hit

And the Side B has been wiped out - while Side A has barely had a quarter of it's strenght taken out.
I'd take a victory like that any day of the week.